4 views

Skip to first unread message

Jul 15, 2019, 6:18:22 AM7/15/19

to Homotopy Type Theory, construc...@googlegroups.com

Dear all,

Readers of this list may be interested in the following series of papers:

[1] S. Henry, Weak model categories in classical and constructive mathematics,

[2] S. Henry, A constructive account of the Kan-Quillen model structure and of Kan's Ex∞ functor,

[3] N. Gambino, C. Sattler, K. Szumiło, The constructive Kan-Quillen model structure: two new proofs

[4] N. Gambino, S. Henry, Towards a constructive simplicial model of Univalent Foundations

Apologies for cross-posting.

With best regards,

Nicola

Dr Nicola Gambino

Associate Professor in Pure Mathematics

School of Mathematics, University of Leeds

Jul 17, 2019, 1:57:13 PM7/17/19

to Homotopy Type Theory

Thanks for collecting all those links together, Nicola!

One of the aspects of this theory that I find especially interesting

is the observation that many uses of AC in classical model category

theory can be avoided by working with "fibration structures" and

requiring all factorization and lifting "properties" to be instead

given by functions. Of course a similar perspective is present in the

notions of algebraic model category (and algebraic weak factorization

system) that have recently been playing a bigger role even in

classical homotopy theory, so it's interesting that the natural

constructive approach is also to work with structure rather than

properties, even in the "non-algebraic" case when the structure isn't

at all "coherent".

Most of these papers describe the situation with phrases like "we are

working in the internal language of a category with finite limits" or

an elementary topos with NNO, or in CZF, and by an "abuse of language"

we interpret "for all x there exists a y" as referring to the giving

of a function assigning a y to each x. But wouldn't it be more

precise and less abusive to just work in dependent type theory with

Sigma and Id types, and sometimes Pi and Nat, and use the untruncated

propositions-as-types logic where "for all x there exists a y"

literally means Pi(x) Sigma(y) and therefore (by the "type-theoretic

principle of non-choice") automatically induces a function assigning a

y to each x? That would also allow asking and answering the question

of how much UIP is required -- do these model structures exist in

HoTT?

> --

> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.

> To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeThe...@googlegroups.com.

> To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/D3003278-EEC5-46A0-A07A-AD260A830DB2%40leeds.ac.uk.

> For more options, visit https://groups.google.com/d/optout.

One of the aspects of this theory that I find especially interesting

is the observation that many uses of AC in classical model category

theory can be avoided by working with "fibration structures" and

requiring all factorization and lifting "properties" to be instead

given by functions. Of course a similar perspective is present in the

notions of algebraic model category (and algebraic weak factorization

system) that have recently been playing a bigger role even in

classical homotopy theory, so it's interesting that the natural

constructive approach is also to work with structure rather than

properties, even in the "non-algebraic" case when the structure isn't

at all "coherent".

Most of these papers describe the situation with phrases like "we are

working in the internal language of a category with finite limits" or

an elementary topos with NNO, or in CZF, and by an "abuse of language"

we interpret "for all x there exists a y" as referring to the giving

of a function assigning a y to each x. But wouldn't it be more

precise and less abusive to just work in dependent type theory with

Sigma and Id types, and sometimes Pi and Nat, and use the untruncated

propositions-as-types logic where "for all x there exists a y"

literally means Pi(x) Sigma(y) and therefore (by the "type-theoretic

principle of non-choice") automatically induces a function assigning a

y to each x? That would also allow asking and answering the question

of how much UIP is required -- do these model structures exist in

HoTT?

> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.

> To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeThe...@googlegroups.com.

> To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/D3003278-EEC5-46A0-A07A-AD260A830DB2%40leeds.ac.uk.

> For more options, visit https://groups.google.com/d/optout.

Jul 18, 2019, 3:55:41 AM7/18/19

to Michael Shulman, Homotopy Type Theory

Dear Mike,

On 17 Jul 2019, at 18:56, Michael Shulman <shu...@sandiego.edu> wrote:

Most of these papers describe the situation with phrases like "we are

working in the internal language of a category with finite limits" or

an elementary topos with NNO, or in CZF, and by an "abuse of language"

we interpret "for all x there exists a y" as referring to the giving

of a function assigning a y to each x. But wouldn't it be more

precise and less abusive to just work in dependent type theory with

Sigma and Id types, and sometimes Pi and Nat, and use the untruncated

propositions-as-types logic where "for all x there exists a y"

literally means Pi(x) Sigma(y) and therefore (by the "type-theoretic

principle of non-choice") automatically induces a function assigning a

y to each x? That would also allow asking and answering the question

of how much UIP is required -- do these model structures exist in

HoTT?

Thank you for your email.

Your suggestion of working in a dependent type theory is interesting. I am not sure what kind of dependent type theory would be sufficient to develop these papers and what would be the best approach to the formalization (e.g. via sets-as-hsets or via sets-as-setoids).

Regarding the dependent type theory, apart from basic rules, I guess one would need:

- some extensionality,

- propositional truncations,

- pushouts,

- some inductive types (for the instances of the small object argument)

- at least one universe (cf. quantification over all Kan complexes).

One could then keep track explicitly of which existential quantifies are to be left untruncated and which ones can be truncated, and then see if everything can be done in HoTT.

Is this the kind of thing you had in mind?

Another approach to avoiding the abuse of language, suggested by Andre’ Joyal, is to develop a theory of “split” weak factorisation systems, i.e. weak factorisation systems in which one has a given choice of fillers, and work with them. This would be a
variant of the theory of algebraic weak factorisation systems. We are working on that.

With best wishes,

Nicola

PS The first link in my email was incorrect. Simon Henry’s paper "Weak model categories in classical and constructive mathematics” is available at https://arxiv.org/abs/1807.02650.

Jul 18, 2019, 4:16:09 AM7/18/19

to Nicola Gambino, Michael Shulman, Homotopy Type Theory

In our work on GCTT we used the internal DTT/DPL of a topos.

https://arxiv.org/abs/1611.09263 (sec 4)

There's a convenient presentation of this in the work of Phao (appendix 1)

www.lfcs.inf.ed.ac.uk/reports/92/ECS-LFCS-92-208/

and the elephant D4.3,4.4.

It may not give you everything that you need, but it may be a start.

https://arxiv.org/abs/1611.09263 (sec 4)

There's a convenient presentation of this in the work of Phao (appendix 1)

www.lfcs.inf.ed.ac.uk/reports/92/ECS-LFCS-92-208/

and the elephant D4.3,4.4.

It may not give you everything that you need, but it may be a start.

> --

> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.

> To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeThe...@googlegroups.com.

> To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/D49A1FEA-4CE1-448F-97A8-46065AF9E7B6%40leeds.ac.uk.
> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.

> To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeThe...@googlegroups.com.

Jul 18, 2019, 8:21:37 AM7/18/19

to Nicola Gambino, Homotopy Type Theory

For something that ought to work in the internal language of a

category with finite limits, like the first paper on weak model

categories, it should technically suffice to have Sigma- and Id-types

with UIP (since those are a version of that internal language). If we

wanted to internalize more of the abusive external statements like "f

is an acyclic fibration if it has right lifting for every

cofibration", it should be enough to add Pi-types and a universe.

To enhance this to the internal language of an elementary

(predicative) 1-topos with NNO or an analogue of CZF, coproducts,

propositional truncations, pushouts, and a natural numbers type should

be enough. I'm very curious to hear where propositional truncations

and pushouts are used (if ever), since in so many places the "for all

x, exists y" is actually an untruncated Sigma. Certainly pushouts

appear in the small object argument, but I wonder whether those

pushouts could be implemented with coproducts in the case when they

are pushouts of cofibrations that are sufficiently "complemented

inclusions" (the pushout of A -> X along a complemented inclusion A ->

A+B is just X+B).

category with finite limits, like the first paper on weak model

categories, it should technically suffice to have Sigma- and Id-types

with UIP (since those are a version of that internal language). If we

wanted to internalize more of the abusive external statements like "f

is an acyclic fibration if it has right lifting for every

cofibration", it should be enough to add Pi-types and a universe.

To enhance this to the internal language of an elementary

(predicative) 1-topos with NNO or an analogue of CZF, coproducts,

propositional truncations, pushouts, and a natural numbers type should

be enough. I'm very curious to hear where propositional truncations

and pushouts are used (if ever), since in so many places the "for all

x, exists y" is actually an untruncated Sigma. Certainly pushouts

appear in the small object argument, but I wonder whether those

pushouts could be implemented with coproducts in the case when they

are pushouts of cofibrations that are sufficiently "complemented

inclusions" (the pushout of A -> X along a complemented inclusion A ->

A+B is just X+B).

> --

> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.

> To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeThe...@googlegroups.com.

> To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/D49A1FEA-4CE1-448F-97A8-46065AF9E7B6%40leeds.ac.uk.
> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.

> To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeThe...@googlegroups.com.

Reply all

Reply to author

Forward

0 new messages

Search

Clear search

Close search

Google apps

Main menu